In a word: No.
I predict with great confidence that we'll not see a two-thirds majority in both houses on this subject. Remember, the Congress is so deeply split that it can barely agree to hold committee hearings, let alone pass amendments to the basic law of the land. And even if somehow such an amendment gets out of the Congress, there's no way that 38 states would ratify it. Oh, sure, there would be a rush as maybe as many as twenty or so state legislatures ran to ratify, but you'll never get three-fourths of the states to agree to it.
Constitutions are supposed to be difficult to change. That's one reason that California's governance is in a horrible mess. It's too easy to push constitutional amendments through the initiative process. Thus California's state constitution is a monster document full of messy contradictions and too many patches and amendments. Kudos to those states whose people adopted amendment systems that take a long time and multiple legislative sessions, so that passions of the moment can cool and demagogues can lose momentum.
For those people screaming "judicial activism!" (="The ruling didn't go my way!"), I say this: Fundamental civil rights should not be put to a public vote. If we did that, slavery would still be the law of the land in many states, anti-miscegenation laws would still be in force, and I expect that some states would have religious test laws in place that banned anyone except those with certain religious affiliations from even living in their state, let alone holding office. And for all I know, I'd be living on a reservation (and prohibited from leaving), given that I'm about 3/8 American Indian*, and thus not a proper White Man.
Incidentally, as annoyed as I am by various legislators calling for the repeal of the 14th Amendment to deal with the Brown Peril, I think their chance of success is just as low as that of those who want a Marriage Amendment.
A somewhat more achievable medium-term goal of the Cultural Conservatives might be to try again to split the Ninth Circuit Court, in an attempt to limit the "spread" of decisions coming out of California, especially if the Supreme Court decides not to take the case when it gets that far. Of the various proposals listed in that Wikipedia article, the one that would redefine the Ninth Circuit as California, Guam, Hawaii, and the Northern Mariana Islands, putting the rest of the states into a new Twelfth Circut, seems most logical to me. But I digress into political geography.
*I don't meet the "culturally identify" portion of the Census Bureau definition of American Indian. I note my Cherokee-Chippewa ancestry in the same way that I have German, English, and Irish ancestors of which I'm aware.